SciPost Submission Page
Hamiltonian structure of 2D fluid dynamics with broken parity
by Gustavo Machado Monteiro, Alexander G. Abanov, Sriram Ganeshan
This is not the latest submitted version.
|As Contributors:||Sriram Ganeshan · Gustavo Machado Monteiro|
|Arxiv Link:||https://arxiv.org/abs/2105.01655v2 (pdf)|
|Date submitted:||2021-08-17 17:06|
|Submitted by:||Machado Monteiro, Gustavo|
|Submitted to:||SciPost Physics|
Isotropic fluids in two spatial dimensions can break parity symmetry and sustain transverse stresses which do not lead to dissipation. Corresponding transport coefficients include odd viscosity, odd torque, and odd pressure. We consider an isotropic Galilean invariant fluid dynamics in the adiabatic regime with momentum and particle density conservation. We find conditions on transport coefficients that correspond to dissipationless and separately to Hamiltonian fluid dynamics. The restriction on the transport coefficients will help identify what kind of hydrodynamics can be obtained by coarse-graining a microscopic Hamiltonian system. Interestingly, not all parity-breaking transport coefficients lead to energy conservation and, generally, the fluid dynamics is energy conserving but not Hamiltonian. We show how this dynamics can be realized by imposing a nonholonomic constraint on the Hamiltonian system.
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Anonymous Report 2 on 2022-2-18 (Invited Report)
The paper considers non-relativistic parity violating fluids in 2 spatial dimensions, and asks, whether the non-dissipative part of the system can be described by Hamiltonian dynamics. In general, a non-relativistic fluid is described by a stress-energy-momentum complex. The authors parameterize the stress tensor in terms of viscosity coefficients, and constrain them by Galilean invariance. The question then is when the dynamical system has a conserved energy operator. The authors address this by carrying out a classical phase space analysis. The discussion is interesting and helps understand some of the structure of parity-odd terms in 2 spatial dimensions. The paper is also clearly written. I recommend it be accepted for publication.
Anonymous Report 1 on 2021-11-23 (Invited Report)
The authors identify new conditions on odd parity transport coefficients in 2D for the resulting hydrodynamic equations to be (i) energy conserving (ii) Hamiltonian. The results are very interesting and should certainly be published in Scipost. There were a few points that I would like the authors to clarify before publication:
1) In determining when these systems are Hamiltonian (the criterion for energy conservation is reasonably unambiguous), the authors assume that (i) the Hamiltonian is the energy (ii) the brackets take the specific form defined by Eqs. (18), (19), (26). Within these assumptions, they derive conditions on the transport coefficients for the system to be Hamiltonian. As far as I can tell, these are sufficient conditions – can the authors rule out the possibility of a different Hamiltonian (i.e. not necessarily the energy) and different Poisson brackets whenever the conditions of Statement III are not met? Or does the authors’ physical interpretation of Statement II show that it captures all possible Hamiltonian cases? As things are written, the claim seems to be a bit vague, e.g. on p3, “we will address when the energy conserving fluid dynamics described in Statement I can be endowed with the Hamiltonian structure”, “we show that not all cases in Statement I can possess Hamiltonian structure”, and Statement III makes no reference to a particular choice of bracket.
2) The authors mention in the abstract that their results constrain odd-parity hydrodynamics that derives from coarse-grained microscopic systems. It seems to me that this is only true if Eqs. (18), (19), (26) also derive from coarse-graining the microscopic canonical brackets in some way – can this be justified? It seems plausible for Eqs. (18) and (19) but less so for Eq. (26). If not, there is the risk that being Hamiltonian is a formal property of the macroscopic hydrodynamics that is unrelated to the microscopic dynamics.
3) Is there any physical significance to the fact that the “counting scheme” on p6 differs from the usual hydrodynamic derivative expansion? Naively this mixing of different orders in the derivative expansion seems difficult to reconcile with hydrodynamic scaling (e.g. the “Madelung terms” are dispersive and usually neglected in low-order hydrodynamics)
4) The explicit connection to nonholonomic constraints in Sec. IV is nice. Is this expected in all cases that violate Statement III? (there seems to be a claim in the literature that violating Jacobi is always equivalent to a nonholonomic constraint in finite dimensions, e.g. Van Der Schaft, Maschke, Rep. Math. Phys. 34 2 p.225-233, 1994). The way things are currently stated in Sec. IV seems possibly overly general: “In this section, we consider the fluid dynamics described by Case 1 of Statement I, but not satisfying the condition of Statement II. We show that it arises from Hamiltonian fluid dynamics with internal angular momentum degree of freedom subject to a nonholonomic constraint”, given that it is illustrated with one specific example.
5) The conclusion felt a little technical and confusing, with references back to very specific details and some new details added (e.g. central extensions). Please consider revising.
1) I found v_i^2 in Eqs. 14-16 a bit confusing, similarly j_i^2 elsewhere. Could the authors consider an alternative notation?
2) Appendices: what (four times) -> which